Phase-Locked Signals Elucidate Circuit Architecture of an Oscillatory Pathway

This paper introduces the concept of phase-locking analysis of oscillatory cellular signaling systems to elucidate biochemical circuit architecture. Phase-locking is a physical phenomenon that refers to a response mode in which system output is synchronized to a periodic stimulus; in some instances, the number of responses can be fewer than the number of inputs, indicative of skipped beats. While the observation of phase-locking alone is largely independent of detailed mechanism, we find that the properties of phase-locking are useful for discriminating circuit architectures because they reflect not only the activation but also the recovery characteristics of biochemical circuits. Here, this principle is demonstrated for analysis of a G-protein coupled receptor system, the M3 muscarinic receptor-calcium signaling pathway, using microfluidic-mediated periodic chemical stimulation of the M3 receptor with carbachol and real-time imaging of resulting calcium transients. Using this approach we uncovered the potential importance of basal IP3 production, a finding that has important implications on calcium response fidelity to periodic stimulation. Based upon our analysis, we also negated the notion that the Gq-PLC interaction is switch-like, which has a strong influence upon how extracellular signals are filtered and interpreted downstream. Phase-locking analysis is a new and useful tool for model revision and mechanism elucidation; the method complements conventional genetic and chemical tools for analysis of cellular signaling circuitry and should be broadly applicable to other oscillatory pathways.

: Representative cases of individual intracellular calcium signals vs. time graphs for cells addressed with periodic carbachol stimulation (C = 10 nM, D = 24 s, R = 24 s). I/I 0 is the normalized FRET ratio of the intracellular calcium signals, as was used in Fig. 1. The '(-)' symbols denote skipped beats, based upon the criteria explained in detail in the Materials and Methods Section. Sub-threshold spikes are apparent in a majority of the graphs. A greater proportion of cells exhibit skipped beats compared to Text S1 Fig. 1, since the stimulant concentration is lower.    Chay et al. model (red), and the former model with basal PLC activity (blue). The times and magnitudes of the IP3 curves have been offset for easier comparison. For each curve, the stimulation duration was 10 s and the rest period was 500 s. The stimulant concentration was 0.05 1/s for the Chay et al. model, 1.2 μM/s for the Politi et al. model, and 0.3 μM/s for the latter model with basal IP3 production. Basal IP3 production was set at 0.3 μM/s. models supplemented with ligand, receptor, and G-protein dynamics. a) Calcium oscillation period vs. C (stimulant concentration). b) Phase-locking ratio vs. C, with D = 25 s and R = 25 s. c) Phase-locking ratio vs. D, with C = 55 nM (left), 15 nM (right) and R = 25 s. d) Phase-locking ratio vs. R, with C = 55 nM (left), 15 nM (right) and D = 25 s. e) Individual calcium signal vs. time graphs, with C = 55 nM (left), 15 nM (right), D = 25 s, and R = 25 s. Parameter values and rate equations for the ligand, receptor, G-protein dynamics were taken from [1,2]. Notably, the behaviors of the Chay et al. and Politi et al. models with enhanced biochemical detail under periodic stimulation are similar to the original models ( Fig. 3 and Text S1 Fig. 3). Fig. 7: Phase-locking analysis of the Cuthbertson and Chay model. a) Calcium oscillation period vs. C (stimulant concentration). b) Phase-locking ratio vs. C, with D = 25 s and R = 25 s. c) Phase-locking ratio vs. D, with C = 0.015 1/s and R = 25 s. d) Phase-locking ratio vs. R, with C = 0.015 1/s and D = 25 s. e) Individual calcium signal vs. time graph with the following periodic stimulation parameters: C = 0.015 1/s, D = 25 s, and R = 25 s. In d), the phase-locking ratio decreases for increases in rest period, and in e), there is an absence of sub-threshold spikes. b) Phase-locking ratio vs. C, with D = 10 s and R = 3 s. c) Phase-locking ratio vs. D, with C = 0.57 and R = 3 s. d) Phase-locking ratio vs. R, with C = 0.57 and D = 10 s. e) Individual calcium signals vs. time graph, with the following periodic stimulation parameters: C = 0.57, D = 10 s, and R = 3 s. The model predicts all the correct behaviors seen experimentally under periodic stimulation, with the caveat that the calcium oscillation dynamics are much faster than what was observed experimentally. Fig. 9: Phase-locking analysis of the Li and Rinzel Model (a version adapted for the study by Sneyd et al. [3]). a) Calcium oscillation period vs. C (stimulant concentration). b) Phase-locking ratio vs. C, with D = 20 s and R = 20 s. c) Phase-locking ratio vs. D, with C = 0.4 µM/s and R = 20 s. d) Phase-locking ratio vs. R, with C = 0.4 µM/s and D = 20 s. e) Individual calcium signal vs. time graph, with the following periodic stimulation parameters: C = 0.4, µM/s D = 20 s, R = 20 s. In b) and c), the phase-locking ratio decreases with increases in C and D, opposite of what was observed experimentally. , the phase-locking ratio remains constant with increases in C, which was not observed experimentally; in d), the phaselocking ratio decreases to zero and then increases with increases in R, which was also not observed experimentally. Fig. 12: Phase-locking analysis of two models of Circadian rhythms: the Tyson et al. model [4] and the Goldbeter model [5]. a) Plotting the phase-locking ratio vs. R for the Tyson et al. model revealed that for small stimulation durations (D), the phase-locking ratio increased, then decreased to zero. The stimulation parameters used to generate this graph were: C = 1 Cm/hr and D = 2 hrs. For larger D, it was found that the phase-locking ratio increased, and remained at the value one, suggesting that the recovery properties of the Tyson et al. model depend partly on D. b) For the Goldbeter model, an increase in R resulted in a corresponding increase in the phase-locking ratio for both small and large D. These results suggest that the recovery properties of the Goldbeter model do not depend on D.
The following stimulation parameters were used to generate the graph depicted: C = 2 μM/hr and D = 2 hrs.

Mathematical Modeling
The following section contains all the model equations, parameters, and initial conditions for the 9 mathematical models of oscillatory calcium signaling analyzed in this study; also included are brief descriptions of each model.
Additional model details can be found in the original publications; references are provided to direct the reader to these works. The model equations, parameters, and initial conditions for the two circadian models analyzed in this study are also provided in this section.

A. Chay et al. model (Reference [6])
i. ii. Model equations: Rate equations for the ligand/receptor dynamics used for Text S1 Fig. 6: The equation describing activated G-protein dynamics was changed to the following:

B. Politi et al. model (Reference [7])
i. Model Description: The Politi et al. is able to produce calcium oscillation periods on the order of seconds to minutes, similar to what was observed experimentally in our studies. The model features calcium feedback upon IP3 metabolism, a key feature that was found to expand the range of oscillation periods. Oscillations in this model are produced by a feedback scheme whereby IP3 results in calcium release, and calcium then enhances its own release and IP3 production, but at high concentrations, calcium inhibits its own release. Several features of the model were experimentally validated.

E. Dupont et al. model (Reference [11])
i. Model description: The Dupont et al. model is based upon experimental observations that IP3 metabolism, specifically from IP3-3 kinase and IP3-5 phosphatase, significantly affect calcium signaling dynamics. In this model, external stimulation leads to IP3 release, which then results in calcium release. Calcium then engages the two aforementioned enzymes, which result in reduced IP3 levels. This feedback mechanism results in calcium oscillations, and is able to reproduce several experimentally observed calcium signaling behaviors.
Model equations: